Radially symmetric transition-layer solutions in mass-conserving reaction-diffusion systems with bistable nonlinearity

TL;DR

This paper proves the existence of radially symmetric transition-layer solutions in mass-conserving reaction-diffusion systems with bistable nonlinearities, providing a rigorous mathematical framework for pattern formation in biological and ecological models.

Contribution

It introduces a refined asymptotic analysis method that incorporates mass constraints, enabling high-order approximations and error estimates for transition-layer solutions in N-dimensional domains.

Findings

Existence of radially symmetric stationary solutions with a single internal layer

Development of a high-order asymptotic approximation framework

Quantitative error estimates for solutions in arbitrary dimensions

Abstract

Mass-conserving reaction-diffusion (MCRD) systems are widely used to model phase separation and pattern formation in cell polarity, biomolecular condensates, and ecological systems. Numerical simulations and formal asymptotic analysis suggest that such models can support stationary patterns with sharp internal interfaces. In this work, we establish for a general class of bistable MCRD systems the existence of nonconstant radially symmetric stationary solutions with a single internal transition layer on an $N$-dimensional ball, for general spatial dimension $N$. Our approach incorporates the global mass constraint directly into a refined matched-asymptotic framework complemented by a uniform spectral/linear analysis. Beyond mere existence, our framework yields arbitrarily high-order asymptotic approximations of the constructed solutions together with quantitative uniform error estimates, which provides a quantitative higher-dimensional theory of transition-layer patterns in MCRD systems and a rigorous justification for their use in modeling phase separation and pattern formation in biological and ecological settings.